Blow up of solutions of semilinear heat equations in general domains

Abstract

Consider the nonlinear heat equation $v_t -\Delta v= |v|^{p-1} v$ in a bounded smooth domain $\Omega\subset \R^n$ with $n>2$ and Dirichlet boundary condition. Given $u_{p}$ a sign-changing stationary solution fulfilling suitable assumptions, we prove that the solution with initial value $\theta u_{p} $ blows up in finite time if $ |\theta -1|>0$ is sufficiently small and if $p$ is sufficiently close to the critical exponent. Since for $\theta=1$ the solution is global, this shows that, in general, the set of the initial data for which the solution is global is not star-shaped. This phenomenon had been previously observed in the case when the domain is a ball and the stationary solution is radially symmetric.